Benoit Fresse: Rational homotopy and intrinsic formality of E_n-operads

ABSTRACT: The theory of \(E_n\)-operads has considerably developed since a decade. Let us mention, among other applications, the second generation of proofs of the Kontsevich formality theorem, based on the formality of \(E_2\)-operads, which has hinted the existence of an action of the Grothendieck-Teichmüller group on moduli spaces of deformation-quantization of Poisson algebras, and the description of the Goodwillie-Weiss approximations of embedding spaces in terms of functions spaces on structures associated to \(E_n\)-operads.

The main purpose of this talk is to explain an intrinsic formality statement which gives a characterization of \(E_n\)-operads in terms of their cohomology. Recall that the homology of an \(E_n\)-operad is identified with the operad governing graded Poisson algebra structures of degree \(n-1\). We can apply the Sullivan realization functor to the dual object of this operad of Poisson algebras of degree \(n-1\) in order to retrieve an operad in topological spaces.

The intrinsic formality theorem reads as follows: "If an operad in topological spaces has the same rational homology as an \(E_n\)-operad, for some \(n\geq 3\), and is additionally equipped with an involutive isomorphism that mimes the action of a hyperplane reflection in the case \(n|4\), then this operad is rationally weakly-equivalent to the operad in topological spaces which we canonically associate to the operad of Poisson algebras of degree \(n-1\)."

The proof relies on methods of obstruction theory and on the interpretation of obstruction cycles in terms of the homology of certain graph-complexes. To conclude the talk, I will explain that these graph-complexes also determine the rational homotopy type of mapping spaces between \(E_n\)-operads.

This talk is based on joint works with Victor Turchin and Thomas Willwacher.